On sets of convergence and divergence of multiple orthogonal series

M. I. Dyachenko ; K. S. Kazarian

Sbornik. Mathematics, Tome 193 (2002) no. 9, pp. 1281-1301

Résumé

Multiple Fourier series with respect to uniformly bounded orthonormal systems (ONSs) are studied. The following results are obtained. \medskip Theorem 1. \textit{Let $\Phi=\{\varphi_n(x)\}_{n=1}^\infty$ be a complete orthonormal system on $[0,1]$ that is uniformly bounded by $M$ on this interval, assume that $m\geqslant2$, and let $\Phi(m)=\{\varphi_{\mathbf n}(\mathbf x)\}_{\mathbf n \in\mathbb N^m}$, where $\varphi_{\mathbf n}(\mathbf n) =\varphi_{n_1}(x_1)\dotsb\varphi_{n_m}(x_m)$. Then there exists a function $f(\mathbf x)\in L([0,1]^m)$ cubically diverges on some measurable subset $\mathscr H$ of $[0,1]^m$ with $\mu_m(\mathscr H)\geqslant 1-(1-1/M^2)^m$. } \medskip Theorem 3. For $M>1$ and an integer $m\geqslant 2$ let $E$ be an arbitrary measurable subset of $[0,1]$ such that $\mu(E)=1-1/M^2$. Then there exists a complete orthonormal system $\Phi$ on $[0,1]$ uniformly bounded by $M$ there such that the multiple Fourier series of each function $f(\mathbf x)\in L([0,1]^m)$ with respect to the product system $\Phi(m)$ cubically converges to $f(\mathbf x)$ a.e. on $E^m$. \medskip Definitive results in this direction are established also for incomplete uniformly bounded ONSs.